Vector field ( Definition ). The Gauss' theorem. Flux of the vector field through a closed...

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Vector field ( Definition ). The Gauss' theorem. Flux of the vector field through a closed surface ( Calculation, applications, physical meaning ). Task: Find a flux of the vector field a=(2x+z)i+(x+2y)j+(4y+e")k Jz² = 4(x + y) through the closed surface S: 2 = 2 4xy 2. Given the scalar field u(x,y, z)= In(x -3)- and the point M(4, 2, -1). Find the direction, with a start point M, along which the rate of change of the scalar field u is the greatest, and calculate its value. p+1 p(p -10p+50) 3. Find the original of Laplace transform: F(p)= 4. Evaluate the complex line integral (zi+2z)dz, if L is the line y = xr connecting the point z, 1+i with the point z, = 0. 5. Using Stokes formula, find the circulation of the vector field a = 4yi +8xj + zyk along a triangle contour ABC, if A(1,0,0), B(0,3,0), C(0,0, 1). 6. Find the real and imaginary part of the function f(z) = (z+1)ee Check out if the Cauchy-Riemann equations are satisfied, and calculate f'(z) if it exists. Vector field ( Definition ). The Gauss' theorem. Flux of the vector field through a closed surface ( Calculation, applications, physical meaning ). Task: Find a flux of the vector field a=(2x+z)i+(x+2y)j+(4y+e")k Jz² = 4(x + y) through the closed surface S: 2 = 2 4xy 2. Given the scalar field u(x,y, z)= In(x -3)- and the point M(4, 2, -1). Find the direction, with a start point M, along which the rate of change of the scalar field u is the greatest, and calculate its value. p+1 p(p -10p+50) 3. Find the original of Laplace transform: F(p)= 4. Evaluate the complex line integral (zi+2z)dz, if L is the line y = xr connecting the point z, 1+i with the point z, = 0. 5. Using Stokes formula, find the circulation of the vector field a = 4yi +8xj + zyk along a triangle contour ABC, if A(1,0,0), B(0,3,0), C(0,0, 1). 6. Find the real and imaginary part of the function f(z) = (z+1)ee Check out if the Cauchy-Riemann equations are satisfied, and calculate f'(z) if it exists.